Performance Analysis of Ultra Wideband Receivers for High Data Rate Wireless Personal Area Network System

Bikramaditya Das

*

and Susmita Das,

Member

, IEEE

*

*

Department of Electrical Engineering, National Institute of Technology, Rourkela -769008, India

adibik09@gmail.com

sdas@nitrkl.ac.in

A

BSTRACT

For high data rate ultra wideband communication system, performance comparison of Rake, MMSE and Rake-MMSE receivers is attempted in this paper. Further a detail study on Rake-MMSE time domain equalizers is carried out taking into account all the important parameters such as the effect of the number of Rake fingers and equalizer taps on the error rate performance. This receiver combats inter-symbol interference by taking advantages of both the Rake and equalizer structure. The bit error rate performances are investigated using MATLAB simulation on IEEE 802.15.3a defined UWB channel models. Simulation results show that the bit error rate probability of Rake-MMSE receiver is much better than Rake receiver and MMSE equalizer. Study on non-line of sight indoor channel models illustrates that bit error rate performance of Rake-MMSE (both LE and DFE) improves for CM3 model with smaller spread compared to CM4 channel model. It is indicated that for a MMSE equalizer operating at low to medium SNR values, the number of Rake fingers is the dominant factor to improve system performance, while at high SNR values the number of equalizer taps plays a more significant role in reducing the error rate.

K

EYWORDS

UWB, Rake receiver, MMSE, Rake-MMSE, Bit Error Rate, LE, DFE.

1.

I

NTRODUCTION

to study time equalization with combined Rake-MMSE equalizer structure. We show that, for a MMSE equalizer operating at low to medium SNR’s, the number of Rake fingers is the dominant factor to improve system performance, while, at high SNR’s the number of equalizer taps plays a more significant role in reducing error rates[7-8]. We show that for high frequency selective channels such as the CM4 one, a linear equalizer structure is not sufficient and must be replaced by a decision feedback equalizer (DFE) structure. Furthermore, we propose a simple recursive gradient based algorithm to implement the equalizer structures.

The rest of the paper is organized as follows. In Section 2 we study the signals and system model for IEEE UWB channel modelling. Section 3 is devoted principles of equalizations and receiver structure. In section 4 we study performance analysis for Rake-MMSE receiver. Simulation results are discussed in Section 5. Section 6 concludes the paper.

2.

SIGNALS

AND

SYSTEM

MODEL

For a single user system, the continuous transmitted data stream is written

+∞

−∞ =

− =

k

s T k t p k d t

s() ( ). ( . ) (1)

Where d(k)are stationary uncorrelated BPSK data and Ts is the symbol duration. Throughout this paper we consider the application of a root raised cosine (RRC) transmit filter p(t) with roll-off factor = 0.3. The UWB pulse p(t) has duration Tuwb (Tuwb < Ts ).

The channel models used in this paper are the model proposed by IEEE 802.15.3a Study Group [10]. In the normalized models provided by IEEE 802.15.3a Study Group, different channel characteristics are put together under four channel model scenarios having rms delay spreads ranging from 5 to 26 nsec. For this paper four channel models, derived from the IEEE 802.15 channel modelling working group. In IEEE 802.15.3a working group, the UWB channel is further classified into four models. Channel model 1 (CM1) represents LOS and distance from 0 to 4 m UWB channel, while channel model 2 (CM2) represents NLOS and distance from 0 to 4 m UWB channel. Distance from 4 m to 10 m and NLOS UWB channel is modelled as CM3 and distance over 10 m NLOS UWB channels are all classified into the extreme model CM4.

The impulse response can be written as

=

−

=

M

p

p

p

t

h

t

h

0

)

(

.

)

(

δ

τ

(2) Parameter M is the total number of paths in the channel.

3.

PRINCIPLE

OF

RECEIVERS

STRUCTURE

3.1. RAKE RECEIVER

Figure.1. UWB RAKE receiver structure

The received signal first passes through the receiver filter matched to the transmitted pulse and is given by

) ( ) . ( . ) ( ) ( * ) ( ) ( * ) ( * ) ( ) ( t n T k t m h k d t p t n t p t h t s t r i s k i i Λ ∞ + −∞ = + − − = − + − =

τ (3)

Wherep(-t) represents the receiver matched filter, “*” stands for convolution operation and

n(t) is the additive white Gaussian noise (AWGN) with zero mean and variance N0/ 2 . Also,

m (t ) = p (t ) * p ( −t ) and n (t ) = n (t ) * p ( −t ) . Combining the channel impulse response (CIR) with the transmitter pulse shape and the

matched filter, we have

() ()* ()* ( ) . ( ) 0 ~ i M i im t

h t p t h t p t

h = − = −τ

=

(4) The output of the receiver filter is sampled at each Rake finger. The minimum Rake finger

separation is Tm = Ts / Nu , whereNuis chosen as the largest integer value that would result in

Tm spaced uncorrelated noise samples at the Rake fingers

v

(

nT t

)

h

(

(

n k

)

Ts i t

)

d

( )

k k

l

s . .

. 0 ' ~ 0 ' + + − = + + +∞ −∞ = τ

τ (5) where l is the delay time corresponding to thelthRake finger and is an integer multiple ofTm.

Parameter t0corresponds to a time offset and is used to obtain the best sampling time. Without

loss of generality,t0will be set to zero in the following analysis.

3.2. MMSE STRUCTURE

In reality the noise component due to the physical channel cannot be ignored. In the presence of additive Gaussian noise at the receiver input, the output of the equalizer at the nth sampling instant is given by

− = − = N N K k n k

n b r

yˆ (6) The mean square error (MSE) for the equalizer having 2N+1 taps, denoted by J(N) is

_{= − = −} − = − 2 2 ˆ ) ( N N k k n k n n

n y E x b r

x E N

J (7)

J(N)with respect to the equalizer coefficients (bk) is obtained by the following differentiation:

( ) = 0

k

db N dJ

(8)

R_rb = R_xr b = R_r−1R_xr (9)

WHERE

R_xr =(R_xr(−N)....R_xr(0)....R_xr(N))T (10)

) 0 ( ) ( ) 2 (

) 1 ( ) 1 ( ) 1 2 (

. . .

. . .

) 2 ( )... ( )... 0 (

2 −

−

+ − + − =

r r

N r

r r

r

r r

r

r

R N

R N R

R N

R N R

N R N

R R

R (11)

3.3. RAKE-MMSE STRUCTURE

The receiver structure is illustrated in Fig. 2 and consists in a Rake receiver followed by a linear equalizer. As we will see later on, a structure gives better performances over UWB channels when the number of equalizer taps is sufficiently large. The received signal first passes through the receiver filter matched to the transmitted pulse (3). The output of the receiver filter is sampled at each Rake finger. The minimum Rake finger separation is Tm = Ts / Nu , where Nu is chosen as the largest integer value that would result in Tm spaced uncorrelated noise samples at the Rake fingers(4). In a first approach, complete channel state information (CSI) is assumed to be available at the receiver. For general selection combining, the Rake fingers ( ’s) are selected as the largest L (L Nu) sampled signal at

Figure.2. UWB RAKE-MMSE receiver structure

the matched filter output within one symbol time period at time instants l ,l = 1, 2, ..., L . A

feasible implementation of multipath diversity combining can be obtained by a selective-Rake (SRake) receiver, which combines the L best, out of Nu, multipath components. Those L best components are determined by a finger selection algorithm. For a maximal ratio combining (MRC) Rake receiver, the paths with highest signal-to-noise ratios (SNRs) are selected, which is an optimal scheme in the absence of interfering users and intersymbol interference (ISI). For a minimum mean square error (MMSE) Rake receiver, the “conventional” finger selection algorithm is to choose the paths with highest signal-to-interference-plus-noise ratios (SINRs) [2]. The received signal sampled at the lth Rake finger in the nth data symbol interval is given by equation (5).

The Rake combiner output at time t = n.Ts is

(

)

(

'

)

l s L

1 l

l '

l s L

1 l

l.v n.T .n n.T

] n [

y = β +τ + β Λ +τ =

=

(12)

4.

PERFORMANCE

ANALYSIS

In this part, due to the lack of place we will only discuss the matrix block computation of linear equalizers. Furthermore, we suppose perfect channel state information (CSI). Assuming that the n data bit is being detected, the MMSE criterion consists in minimizing

) ( ) ( 2 − dΛ n n d E (13)

where d( n ) is the equalizer output. Rewriting the Rake output signal, one can distinguish the desired signal, the undesired ISI and the noise as

(

( )

)

(

'

)

l s L 1 l l ' l s ~ n k L 1 l l ' l ~ L 1 l l T . n n . ) k ( d . T . k n h . ) n ( d . ) ( h . ) n ( y τ β τ β τ β Λ + + + − + = = ≠ =

= (14)

where the first term is the desired output. The noise samples at different fingers,n (n.Ts + l), l

= 1... L, are uncorrelated and therefore independent, since the samples are taken at approximately the multiples of the inverse of the matched filter bandwidth. It is assumed that the channel has a length of (n1 + n2 + 1).Ts. That is, there is pre-cursor ISI from the subsequent

n1 symbols and post-cursor ISI from the previous n2 symbols, and n1 and n2 are chosen large enough to include the majority of the ISI effect. Using (8), the Rake output can be expressed now in a simple form as

( ) ( ) ( )

[ ]n n ( )n d n n k n d n d n y T n k n k k ~ ~ 0

0. .

) ( 2 1 + = + − + = ≠− = φ α α (15)

where coefficient K’s are obtained by matching (14) and (15).

T 2 1

n

n and d[n] = [d(n + n )....d(n). ...d(n - n )]

= [ ... ... ] 2

1 α 0 α

α φ

The superscript denotes the transpose operation. The output of the linear equalizer is obtained as

d n c y

₍

n r

₎

cT

_{( )}

n cT

_{( )}

n

k k r

r − = γ + η

= − = Λ 2 1 . )

( (16) wherec = [c−K1 ...c0 ...c K 2]contains the equalizer taps. Also

[ ]

[

[

]

[ ]

[

]

]

T T T T K n d n d K n d

n =φ + 1...φ ...φ − 2

γ

[ ]

(

)

( )

(

)

T K n n n n K n n

n = + = 2

~ ~ 1 ~ ... ... η (17) The mean square error (MSE) of the equalizer,

_{E d}

_{( )}

_{n − c}T_γ

_{[ ]}

_{n −c}T_η

_{[ ]}

_n 2 (18)

which is a quadratic function of the vector c, has a unique minimum solution. Here, the expectation is taken with respect to the data symbols and the noise. Defining matrices R, p and

N as

p = E

[

d

( )

n .

γ

[ ]

n

]

(20)

_{N =} E

[

_η

[ ]

n ._ηT

[ ]

n

]

(21)

The equalizer taps are given by

c

=

(

R

+

N

)

−1

.

p

(22) and the MMSE is

J

p

R

N

p

T

d

(

)

.

1 2 min −

+

−

=

σ

₍₂₃₎

[ ( ) ] 2 2 n d E d =

σ

Evaluating the expectation over R and p with respect to the data and the noise, we have

[

0

]

[ ]

, 1, 1

2 1 2 1 2 ... ... ₋ = _{+ + + +} = K K K K j i T K

K R r

p α α α

(24) Where 1 n n , 1 n n k , l j i, j , i T j ,

i F and F [f ] 1 2 1 2

r =

φ

φ

= + + + +

− ≠ − − = − = i j k l i j k l flk , 0 , 1 (25)

[ ]

[ ]

[

]

I . . 2 N n . n E

N K K 1

L 1 l 2 l 0 T 2 1+ + =

=

= η η β

(26)

Where I is the identity matrix. This Rake-equalizer receiver will eliminate ISI as far as the number of equalizer’s taps gives the degree of freedom required. In general, the equalizer output can be expressed as

( )

. ( ) .

(

)

( )

0 1

0d n q d n i w n

q n d i + − + = = Λ (27)

.

with

q

_n

=

α

_n

c

_n

The variance of w( n ) is

_{( )}

2

.

.

0 1 2 1 2 2 2 1

N

E

c

p L l K K i i n w

=

= − =

β

σ

(28)

WhereEpis the pulse energy.

the case of DFE, assuming error free feedback, the input data vector can be written in the form of

[ ] [ [ ]... [ ] [ 1]... ] 2

1 n K

T T

DFE n = Φ d n+K Φ d n d n− d −

γ

(29) Using the same approach as for the linear equalizer, the MMSE feedforward taps for tap

equalizer are obtained as

CDFE RDFE NDFE pDFE

1

)

( + −

= ₍₃₀₎

Where CDFE =[c−K₁ ....c₀ 0....0]

Also =

Where RF=(K1+1) square matrix with its element defined by (19) .Matrix U is defined by 1 K 1,... ,... 1 ij 1 2

]

u

[

+ = =

=

_{i K}

U

(32)

Matrix NDFE and vector

p

DFE are given by

p [ K ... 0...0 ]

O O O I 2 / N N 0 1 DFE K K 1 K , K L 1 l K , 1 K K 2 l 0 DFE 2 , 2 1 2 2 1 1 1 α α β = = + = + + (33)

Where matrix 0 is the all zero matrix.

The MMSE feedback taps are then obtained in terms of feed forward taps and matrixU.

[c1………….ck2]=[c-K1………c0]UT (34)

Conditioned on a particular channel realization, h= [h1………hI],an upper bound for the

probability of error using the chernoff bound technique given by

₂ ) / 1 exp( ) ˆ ( min 2 min J J h d d P d n n σ − − ≤

≠ ₍₃₅₎

An exact BER expression with independent noise and ISI terms can be expressed as a series expansion is given by

_∏

≠ = = × − − = ≠ 2 1 0 1 0 2 2 ) cos( ) sin( ) 2 / exp( 2 2 1 ) ˆ ( N n N n n z zodd z n n zwq z zwq w z h d d P

π (36)

Note that ISI comes from the interfering symbols in the range of N1Tsand N2Ts. Parameter z and w determine the accuracy of the error rate given by (35). Set the qi’ s that are within the span of the feedback taps to be 0, which corresponds to zero post-cursor ISI for the span of feedback taps.

5.

SIMULATION

STUDY

AND

ANALYSIS

5.1. Signal Waveform

The pulse shape adopted in the numerical calculations and simulations is the second derivative of the Gaussian pulse given by

w(t)=[1-4

_π

(t/

_ε

)2]exp(-2

_π

(t/

_ε

)2) ₍₃₇₎

5.2. Channel Model Parameter (IEEE 802.15.3a)

to this sampling rate, time channel spread is chosen equal to 100 for CM4 and 70 for CM3, this corresponds to respectively 12 =100 / 8 and 9 = 70 / 8 transmitted symbols. This choice enables to gather 99% of the channel energy. The coherence bandwidths of CM3 and CM4 simulation are 10.6 MHz and 5.9 MHz respectively. The data rate is chosen to be 200 Mbps, one of the optional data rates proposed for IEEE standard. The size of the transmitted packets is equal to 2560 BPSK symbols including a training sequence of length 512. CIR remains constant over the time duration of a packet. The root raised cosine (RRC) pulse with roll off factor

α

=0.5 is employed as the pulse-shaping filter. The CM3 and CM4 indoor channel model is adopted in simulation. The simulated channel impulse responses for CM3 and CM4 are shown in figure 4 and figure 6. The power delay profiles for CM3 and CM4 are plotted in figure 5and figure 7 respectively: The simulation parameter settings for the two channel models are listed in Table.1. The pulse waveform and power spectral density are showed as figure 3.

Table.1 Parameter Settings for IEEE UWB Channel Models

Scenario

Λ

(1/ns)

λ

(1/ns)

Γ

(1/ns)

γ

(1/ns)

_σ

_ξ(dB)

σ

ς (dB)

σ

g(dB)

CM3 0.0067 2.1 14 7.9 3.3941 3.3941 3

CM4 0.0067 2.1 24 12 3.3941 3.3941 3

-1 -0.5 0 0.5 1

x 10-9 -0.5

0 0.5 1

TIME

A

M

P

L

I

T

U

D

E

0 2 4 6

x 1010 0

0.05 0.1 0.15 0.2

FREQUENCY

A

M

P

L

I

T

U

D

E

4 6 8 10 12 14 16

x 109 -120

-100 -80 -60 -40 -20 0

FREQUENCY

P

S

D

Figure.3. Second derivative of Gaussian pulse

Figure.5. Power Delay Profile of CM3 Channel model

Figure.6. Channel Impulse Response of CM4 (NLOS)

Figure.7. Power Delay Profile of CM4 Channel model

5.3. BER ANALYSIS

0 2 4 6 8 10 12 14 16 10-5

10-4 10-3 10-2 10-1 100

SNR(dB)

B

E

R

CM3 CHANNEL

RAKE-MMSE rake MMSE

Figure.8. Performance of UWB receivers for CM3 channel model

0 2 4 6 8 10 12 14 16

10-5 10-4 10-3 10-2 10-1 100

SNR(dB)

B

E

R

CM4 CHANNEL

RAKE-MMSE MMSE rake

Figure.9. Performance of UWB receivers for CM4 channel model

0 5 10 15 20 25 30 10-4

10-3 10-2 10-1 100

SNR(dB)

B

E

R

K=10,L=10,LE CM3 K=10,L=10,DFE CM3

Figure.10. Performance of UWB rake-MMSE-receiver for different number of equalizer taps and rake fingers for CM3 channel model

0 5 10 15 20 25 30

10-4 10-3 10-2 10-1 100

SNR(dB)

B

E

R

K=20,L=1,LE CM4(upperbound) K=20,L=1,LE CM4

K=3,L=20,LE CM4 K=20,L=10,LE CM4 K=3,L=10,DFE CM4 K=3,L=20,DFE CM4 K=20,L=10,DFE CM4 DFE

LE

Figure.11. Performance of UWB rake-MMSE-receiver for different number of equalizer taps and rake fingers for CM4 channel model

fingers and the equalizer taps are increased simultaneously, i.e. K = 20, L = 10 as shown in figure.11. As expected the receiver has better performance over CM3 with smaller delay spread than CM4. Again BER performance observed on different UWB NLOS channel models (CM3 and CM4) shows that LE fails to perform satisfactorily at high SNR’s due to presence of zeros outside the unit circle. These difficulty BER floor can be overcome DFE structure. A DFE outperforms a linear equalizer of the same filter length, and the performance further improve with increase in number of equalizer taps.DFE performances are computed by Monte-Carlo computer simulations, using a training sequence with length 500.

6.

CONCLUSION

UWB is emerging as a solution for the IEEE 802.15a (TG3a) standard which is to provide a low complexity, low cost, low power consumption and high data-rate among Wireless Personal Area Network (WPAN) devices. For high data rate short range, the receivers combats inter-symbol interference by taking advantage of the Rake and MMSE equalizer structure by using different UWB channel models CM3 and CM4. For a MMSE equalizer operating at low to medium SNR’s, the number of Rake fingers is the dominant factor to improve system performance, while, at high SNR’s the number of equalizer taps plays a more significant role in reducing error rates. One can observe a BER floor at high SNR’s the receiver has better performance over CM3 with smaller delay spread thanCM4 in non-line of sight indoor channel models . Rake-MMSE receiver has around1.8dB performance improvement compared to a channel with rake receivers for CM3 channel model. Again BER performance observed on different UWB channel models (CM3 and CM4) shows that DFE outperforms a LE of the same filter length, and the performance further improve with increase in number of equalizer taps. It is concluded that increasing the number of rake fingers performance become superior at low to medium SNR. These architecture has opened up new directions in designing efficient time domain equalizers for UWB system and can be implemented in DSP processors for real - time applications.

R

EFERENCES

[1] A. Rajeswaran, V.S. Somayazulu and J. Foerster, "Rake performance for a pulse based UWB system in a realistic UWB indoor channel", IEEE ICC’03., vol. 4, pp. 2879-2883, May 2003.

[2] S. Gezici, H. V. Poor and H. Kobayashi, "Optimal and Suboptimal Finger Selection Algorithms for MMSE Rake Receivers in Impulse Radio Ultra-Wideband Systems", in Proc IEEE WCNC 2005, New-Orleans, Mar. 2005.

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Authors

Bikramaditya Das received his B.Tech in Electronics and Telecommunications Engineering from the University of B.P.U.T, Rourkela, Orissa, India, in 2007. From 2008 to till now he is a research Fellow under the Department of Electrical Engineering at the N.I.T, Rourkela, India. He is a Member IEI.